1. Level of Measurement Data is generally represented as numbers, but the numbers do not always have the same meaning and cannot be used in the same way. To distinguish the different ways in which numbers are used, we traditionally have identified the level of measurement of the variables as: nominal, ordinal, interval, and ratio. These levels were developed by Stanley Smith Stevens. Nominal Level Variables For nominal level variables, the numbers are shorthand for the categories of a variable, e.g. – 1 represents married persons, – 2 represents divorced persons, – 3 represents persons who have never been married, etc. The assignment of numbers to the categories is arbitrary and can be changed with no loss of meaning. The only legitimate mathematical operation we can do with nominal level data is count the number of times the different categories appear in the data set. Slide 1

2. Ordinal Level Variables Ordinal level variables are usually associated with labels as well, but the assignment of numbers to the categories is ordered, from low to high, e.g. – 1 is assigned to high school graduates, – 2 is assigned to junior college graduates, – 3 is assigned to college graduates, – 4 is assigned to graduates with a masters degree, etc. or from high to low: – 1 is assigned to graduates with a masters degree, – 2 is assigned to college graduates, – 3 is assigned to junior college graduates, – 4 is assigned to high school graduates, etc. The ordering of the numbers tracks the hierarchy of the labels. Though we can change the actual numbers used, the number assigned each higher level degree must consistently be a bigger number than what was used to represent lower ranked degrees (or a smaller number if ordered from high to low) Slide 2

3. The legitimate mathematical operations we can perform on ordinal data is sorting or ranking, as well as counting. Interval Level Variables Interval level variables have the additional characteristic that the difference between numbers is the same for all possible combinations, e.g.: – the difference between 1 and 2 years of age is the same amount as the difference between 21 and 22 years of age, or 50 and 51, or 65 and 66. – the difference between a height of 60 inches and a height of 55 inches is the same amount of difference as a height of 72 inches and a height of 67 inches. For interval level variables, it is mathematically legitimate to do arithmetic (add, subtract, multiple, and divide) as well as count the values, and sort or rank the values. Ratio Level Variables Ratio level variables have the additional property of having a true zero value so that ratios between values are meaningful, but practically speaking, ratio level data is treated the same as interval level. The commonly cited example is temperature. Slide 3

4. Quantitative and Categorical Variables The distinction between nominal and interval levels of variables is substantial. Computing an average marital status (treating a nominal level variable as interval) does not produce a meaningful result, and can be downright embarrassing. Presenting a count of all of the possible ages of the subjects in a data set (using only the nominal level property of an interval variable) does not communicate as much information as saying the average age was 27.5, with a range from 21 to 57. The differences in the use of data at these levels has led many authors to collapse the number of levels of measurement to two, substituting terms like: – quantitative or metric level instead of interval – categorical, qualitative, or non-metric instead of nominal In practice, ordinal level variable are sometimes treated as quantitative and at other times as categorical. The numeric codes for scale variables (1=disagree, 2=neutral, 3=agree) are generally treated as quantitative data and averaged. The numeric codes for year in school (1=freshman, 2=sophomore, 3=junior, 4=senior) are often not used, and comparisons are made using the categorical labels, e.g. the number (count) of seniors with some characteristic versus the number of juniors with the same characteristic. Slide 4

5. When ordinal level variables are used as quantitative variables, we are emphasizing the rank order of the categories, e.g. 3 ranks higher than 2 or 1, and 2 ranks higher than 1. Since the ranks themselves are interval level data, it is argued that arithmetic on the ordinal values is acceptable. Multiple Variables Measuring the Same Construct The same construct can be represented by variables at different levels of measurement. Education can be represented as – years of school (quantitative), – diploma such as high school, college, or post-graduate (categorical, though we could come up with a numbering scheme that made it quantitative) The implication of these different representations is that we cannot base a correct conclusion on the name of the variable or the construct it represents. A correct understanding of a variable’s level of measurement requires that we look at the numbers in the data set and the coding scheme (numeric codes and labels) applied to the variable. The authors of the text for this course use the labels: quantitative and categorical. We will use their terminology in the first set of homework problems. Slide 5

6. Dichotomous variables Dichotomous (two-category) variables can be used in special ways in statistical analysis. For example, they can be treated as an interval or ordinal level predictor in relationships that call for interval level predictors. Examples of dichotomous variables include: – gender: male, female – marital status: married, not married – support capital punishment: yes, no This will be explored in greater detail later in the course. For now, we will treat dichotomous variables as nominal categories. Slide 6

7. Homework problems This week’s homework problems are on level of measurement. Does the variable have value labels? – If there are no value labels, we will treat it as interval level, it can be used as a quantitative variable, and SPSS would categorize it as Scale. – If there are value labels, the variable is either nominal or ordinal. If the value labels have an order or hierarchy from low to high (or high to low), the variable is ordinal, it can be used either as a quantitative variable or a categorical variable, and SPSS would categorize it as Ordinal. If the value labels do not have an order to them, the variable is nominal, it can only be used as a categorical variable, and SPSS would characterize it as Nominal. Note: – Dichotomous variables should be answered as nominal level variables. – NA is not used as a response in these problems. Slide 7

8. Slide 8 In the SPSS Variable View, the level of measurement can be entered in the Measure column as Nominal, Ordinal, and Scale (for interval level variables), but this does not have much practical significance. Note: all of the variables in my datasets are types as Scale. This MAY or MAY NOT be the correct answer in a homework problem.

9. Slide 9 This problem is a level of measurement homework problem in Moodle. The Notes at the bottom of the problem tell you the data set that is being used, with the name of the variable in SPSS in square brackets. The introductory statement includes the type of problem or the statistical technique to be used, in this example: level of measurement.

10. Slide 10 The first question asks about the level of measurement of the variable for employment status, using Stevens’ framework. The SPSS variable for employment status in GSS2000R is wrkstat.

11. Slide 11 In the Data View of the data editor, we see that wrkstat contains numbers, but we cannot tell whether they are measures or codes.

12. Slide 12 First, to see what labels have been assigned to the variable, we click on the Variable View tab. Second, we look to see what numeric codes are used for missing data for the wrkstat row, in the Missing column. These are values that will not be used in the analysis and labels that we ignore in determining the level of measurement.

13. Slide 13 To see the values have been assigned labels, we look in the Variable View tab, and click in the right side of the cell on the row for wrkstat, in the column called Values.

14. Slide 14 When we clicked on the right end of the cell, the Value Labels dialog box opened. Ignoring the 0 and 9 which were coded as missing data, we see eight entries for work status.

15. Slide 15 Since there are value labels, the variable is either nominal or ordinal. If it might be ordinal, we examine the labels for order. I tried to think of them as describing the amount worked, but that clearly doesn’t work for retired, school, and keeping house. Since I find no plausible order, the variable is nominal rather than ordinal. To close the dialog box, click on the OK button.

16. Slide 16 We select nominal from the drop down list. The next sentence asks us to characterize it using the terminology of the text book. Since it is nominal, it can only be used as a categorical variable.

17. Slide 17 The next sentence asks us to characterize it using SPSS’s terminology. SPSS uses the nominal label for variables that are nominal level in Stevens’ scheme.

18. Slide 18 The next question asks about the level of measurement of the variable for number of hours worked in the past week, using Stevens’ framework. The SPSS variable for number of hours worked in the past week in GSS2000R is hrs1.

19. Slide 19 In the visible rows of the Data View, we see values that range from 38 to 60. Based on the variable label and the data values shown, my initial assessment is that this is an interval level variable.

20. Slide 20 The variable hrs1 uses three numeric codes for missing data: - 1, 98, and 99.

21. Slide 21 The only numbers assigned labels are the codes for missing data. There are no value labels for hrs1. Hrs1 is an interval level variable. Click at the right end of the Values cell for hrs1 to open the Value Labels dialog box.

22. Slide 22 We select interval from the drop down list. The next sentence asks us to characterize the variable using the terminology of the text book. Since it is interval, it qualifies as a quantitative variable.

23. Slide 23 The next sentence asks us to characterize it using SPSS’s terminology. SPSS uses the scale label for variables that are interval level in Stevens’ scheme.

24. Slide 24 The next question asks about the level of measurement of the variable for self-employment, using Stevens’ framework. The SPSS variable for self-employment in GSS2000R is wrkslf.

25. Slide 25 In the Data View, we see very restricted options for values of wrkself: 1, 2, and 9.

26. Slide 26 The variable wrkslf uses three numeric codes for missing data: 0, 8, and 9.

27. Slide 27 If we eliminate the codes for missing data (0, 8, and 9), there are only two valid values (1 and 2). While labels have been assigned to the values for this variable, it has only two categories and we characterize it as a special dichotomous variable. Following the directions for these problems, we choose nominal as the measurement level.

28. Slide 28 The next sentence asks us to characterize the variable using the terminology of the text book. Since it is nominal, it qualifies as a categorical variable.

29. Slide 29 The next sentence asks us to characterize it using SPSS’s terminology. SPSS uses the nominal label for variables that are nominal level in Stevens’ scheme.

30. Slide 30 The next question asks about the level of measurement of the variable for highest academic degree, using Stevens’ framework. The SPSS variable for highest academic degree in GSS2000R is degree.

31. Slide 31 In the Data View, we see restricted options for values of degree: 0, 1, 2, 3, 4, and 9.

32. Slide 32 The variable degree uses three numeric codes for missing data: 7, 8, and 9.

33. Slide 33 If we eliminate the codes for missing data (7, 8, and 9), there are five valid values. The are ordered by level of academic achievement and the number of years it takes to complete the degree. Graduate degrees take more years of school than bachelor degrees, which take more years of school than junior college degrees, etc. There is a clear order to the codes and labels.

34. Slide 34 The next sentence asks us to characterize the variable using the terminology of the text book. Since it is ordinal, it can be used both as a categorical variable and a quantitative variable.

35. Slide 35 The next sentence asks us to characterize it using SPSS’s terminology. SPSS uses the ordinal label for variables that are ordinal level in Stevens’ scheme.

36. Slide 36 We have entered a response for each part of the problem. To grade our answers, we click on the Submit button.

37. Slide 37 The green shading indicates that all of our responses are correct.